| Irreducible free energy expansion and overlaps locking in mean field spin glasses - Adriano Barra We introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlap's fluctuations and, in particular, the linear part of the Ghirlanda-Guerra relationships (GG) (often called Aizenman-Contucci polynomials (AC)) in a very simple way. We show moreover how these constraints are "superimposed" by the symmetry of the model with respect to the restriction required by thermodynamic stability... Downloads: 1 | |

| The mean field Ising model trough interpolating techniques - Adriano Barra Aim of this work is not trying to explore a macroscopic behavior of some recent model in statistical mechanics but showing how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfil our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques... Downloads: 2 | |

| Notes on the polynomial identities in random overlap structures - Peter Sollich In these notes we review first in some detail the concept of random overlap structure (ROSt) applied to fully connected and diluted spin glasses. We then sketch how to write down the general term of the expansion of the energy part from the Boltzmann ROSt (for the Sherrington-Kirkpatrick model) and the corresponding term from the RaMOSt, which is the diluted extension suitable for the Viana-Bray model... Downloads: 1 | |

| A statistical mechanics approach to autopoietic immune networks - Adriano Barra The aim of this work is to try to bridge over theoretical immunology and disordered statistical mechanics. Our long term hope is to contribute to the development of a quantitative theoretical immunology from which practical applications may stem. In order to make theoretical immunology appealing to the statistical physicist audience we are going to work out a research article which, from one side, may hopefully act as a benchmark for future improvements and developments, from the other side, it ... Downloads: 3 | |

| A statistical mechanics approach to Granovetter theory - Adriano Barra In this paper we try to bridge breakthroughs in quantitative sociology/econometrics pioneered during the last decades by Mac Fadden, Brock-Durlauf, Granovetter and Watts-Strogats through introducing a minimal model able to reproduce essentially all the features of social behavior highlighted by these authors. Our model relies on a pairwise Hamiltonian for decision maker interactions which naturally extends the multi-populations approaches by shifting and biasing the pattern definitions of an Hop... Downloads: 2 | |

| A Hebbian approach to complex network generation - Elena Agliari Through a redefinition of patterns in an Hopfield-like model, we introduce and develop an approach to model discrete systems made up of many, interacting components with inner degrees of freedom. Our approach clarifies the intrinsic connection between the kind of interactions among components and the emergent topology describing the system itself; also, it allows to effectively address the statistical mechanics on the resulting networks... Downloads: 3 | |

| A mechanical approach to mean field spin models - Giuseppe Genovese Inspired by the bridge pioneered by Guerra among statistical mechanics on lattice and analytical mechanics on 1+1 continuous Euclidean space-time, we built a self-consistent method to solve for the thermodynamics of mean-field models defined on lattice, whose order parameters self average. We show the whole procedure by analyzing in full details the simplest test case, namely the Curie-Weiss model... Downloads: 4 | |

| Equilibrium statistical mechanics on correlated random graphs - Adriano Barra Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most ... Downloads: 4 | |

| Spin glass polynomial identities from entropic constraints - Peter Sollich The core idea of stochastic stability is that thermodynamic observables must be robust under small (random) perturbations of the quenched Gibbs measure. Combining this idea with the cavity field technique, which aims to measure the free energy increment under addition of a spin to the system, we sketch how to write a stochastic stability approach to diluted mean field spin glasses which explicitly gives overlap constraints as the outcome... Downloads: 3 | |

| A certain class of Curie-Weiss models - Giuseppe Genovese By using a formal analogy between statistical mechanics of mean field spin systems and analytical mechanics of viscous liquids -at first pointed out by Francesco Guerra, then recently developed by the authors- we give the thermodynamic limit of the free energy and the critical behavior of Curie Weiss models for a certain class of generalized spin variables. Then, with the same techniques, we give a complete picture of the bipartite Curie-Weiss model, dealing with the same class of generalized sp... Downloads: 7 | |

| Toward a quantitative approach to migrants integration - Adriano Barra Migration phenomena and all the related issues, like integration of different social groups, are intrinsically complex problems since they strongly depend on several competitive mechanisms as economic factors, cultural differences and many others. By identifying a few essential assumptions, and using the statistical mechanics of complex systems, we propose a novel quantitative approach that provides a minimal theory for those phenomena... Downloads: 5 | |

| Stability properties and probability distribution of multi-overlaps in dilute spin glasses - Adriano Barra We prove that the Aizenman-Contucci relations, well known for fully connected spin glasses, hold in diluted spin glasses as well. We also prove more general constraints in the same spirit for multi-overlaps, systematically confirming and expanding previous results. The strategy we employ makes no use of self-averaging, and allows us to generate hierarchically all such relations within the framework of Random Multi-Overlap Structures... Downloads: 2 | |

| Overlap Fluctuations from Random Overlap Structures - Adriano Barra We investigate overlap fluctuations of the Sherrington-Kirkpatrick mean field spin glass model in the framework of the Random Over- lap Structure (ROSt). The concept of ROSt has been introduced recently by Aizenman and coworkers, who developed a variational approach to the Sherrington-Kirkpatrick model. We propose here an iterative procedure to show that, in the so-called Boltzmann ROSt, Aizenman-Contucci (AC) polynomials naturally arise for almost all values of the inverse temperature (not in a... Downloads: 1 | |

| Critical behavior of random spin systems - Luca De Sanctis We provide a strategy to find in few elementary calculations the critical exponents of the overlaps for dilute spin glasses, in absence of external field. Such a strategy is based on the expansion of a suitably perturbed average of the overlaps, which is used in the formulation of the free energy as the difference between a cavity part and the derivative of the free energy itself, considered as a function of the connectivity of the model... Downloads: 2 | |

| Dilution Robustness for Mean Field Ferromagnets - Adriano Barra In this work we compare two different random dilution of a mean field ferromagnet: the first model is built on a Bernoulli-diluted network while the second lives on a Poisson-diluted network. While it is known that the two models have in the thermodynamic limit the same free energy we investigate on the structural constraints that the two models must fulfill. We rigorously derive for each model the set of identities for the multi-overlaps distribution using different methods for the two dilution... Downloads: 8 | |

| Some thoughts on the ontogenesis in B-cell immune networks - Adriano Barra We are interested in modeling theoretical immunology within a statistical mechanics flavor: focusing on the antigen-independent maturation process of B-cells, in this paper we try to revise the problem of self vs non-self discrimination by mature B lymphocytes. We consider only B lymphocytes: despite this is of course an oversimplification, however such a toy model may help to highlight features of their interactions otherwise shadowed by main driven mechanisms due to i.e... Downloads: 6 | |

| Notes on ferromagnetic diluted P-spin model - Elena Agliari In this paper we develop the interpolating cavity field technique for the mean field ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie-Weiss model to p>2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p=2 models and for p>2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of... Downloads: 10 | |

| Interpolating the Sherrington-Kirkpatrick replica trick - Adriano Barra The interpolation techniques have become, in the past decades, a powerful approach to lighten several properties of spin glasses within a simple mathematical framework. Intrinsically, for their construction, these schemes were naturally implemented into the cavity field technique, or its variants as the stochastic stability or the random overlap structures. However the first and most famous approach to mean field statistical mechanics with quenched disorder is the replica trick... Downloads: 7 | |

| Positive-Overlap Transition and Critical Exponents in Mean Field Spin Glasses - Alessandra Agostini In this paper we obtain two results for the Sherrington-Kirkpatrick (SK) model, and we show that they both emerge from a single approach. First, we prove that the average of the overlap takes positive values when it is non zero. More specificly, the average of the overlap, which is naively expected to take values in the whole interval $[-1,+1]$, becomes positive if we ``first'' apply an external field, so to destroy the gauge invariance of the model, and ``then'' remove it in the thermodynamic l... Downloads: 15 | |

| Equilibrium statistical mechanics of bipartite spin systems - Adriano Barra Aim of this paper is to give an extensive treatment of bipartite mean field spin systems, ordered and disordered: at first, bipartite ferromagnets are investigated, achieving an explicit expression for the free energy trough a new minimax variational principle. Furthermore via the Hamilton-Jacobi technique the same free energy structure is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one... Downloads: 7 | |

| Parameter Evaluation of a Simple Mean-Field Model of Social Interaction - Ignacio Gallo The aim of this work is to implement a statistical mechanics theory of social interaction, generalizing econometric discrete choice models. A class of simple mean field discrete models is introduced and discussed both from the theoretical and phenomenological point of view. We propose a parameter evaluation procedure and test it by fitting our model against three families of data coming from different cases: the estimated interaction parameters are found to have similar positive values establish... Downloads: 9 | |

| Replica symmetry breaking in mean field spin glasses trough Hamilton-Jacobi technique - Adriano Barra During the last years, through the combined effort of the insight, coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean field Sherrington-Kirkpatrick spin glass model have been firmly established. In particular, it has been possible to prove the existence and uniqueness of the infinite volume limit for the free energy, and its Parisi expression, in terms of a variational principle, involving a functional orde... Downloads: 13 | |

| Criticality in diluted ferromagnet - Elena Agliari We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet by analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non trivial phase by commutativity breaking of the infinite volume limit and ... Downloads: 9 | |

| Mean field spin glasses treated with PDE techniques - Adriano Barra Following an original idea of F. Guerra, in this notes we analyze the Sherrington-Kirkpatrick model from different perspectives, all sharing the underlying approach which consists in linking the resolution of the statistical mechanics of the model (e.g. solving for the free energy) to well-known partial differential equation (PDE) problems (in suitable spaces). The plan is then to solve the related PDE using techniques involved in their native field and lastly bringing back the solution in the p... Downloads: 62 | |

| Multitasking network with fast noise - Elena Agliari We consider the multitasking associative network in the low-storage limit and we study its phase diagram with respect to the noise level $T$ and the degree $d$ of dilution in pattern entries. We find that the system is characterized by a rich variety of stable states, among which pure states, parallel retrieval states, hierarchically organized states and symmetric mixtures (remarkably, both even and odd), whose complexity increases as the number of patterns $P$ grows... Downloads: 17 | |

| Mean-field cooperativity in chemical kinetics - Aldo Di Biasio We consider cooperative reactions and we study the effects of the interaction strength among the system components on the reaction rate, hence realizing a connection between microscopic and macroscopic observables. Our approach is based on statistical mechanics models and it is developed analytically via mean-field techniques. First of all, we show that, when the coupling strength is set positive, the model is able to consistently recover all the various cooperative measures previously introduce... Downloads: 3 | |

| Multitasking associative networks - Elena Agliari We introduce a bipartite, diluted and frustrated, network as a sparse restricted Boltzman machine and we show its thermodynamical equivalence to an associative working memory able to retrieve multiple patterns in parallel without falling into spurious states typical of classical neural networks. We focus on systems processing in parallel a finite (up to logarithmic growth in the volume) amount of patterns, mirroring the low-level storage of standard Amit-Gutfreund-Sompolinsky theory... Downloads: 5 | |

| How glassy are neural networks? - Adriano Barra In this paper we continue our investigation on the high storage regime of a neural network with Gaussian patterns. Through an exact mapping between its partition function and one of a bipartite spin glass (whose parties consist of Ising and Gaussian spins respectively), we give a complete control of the whole annealed region. The strategy explored is based on an interpolation between the bipartite system and two independent spin glasses built respectively by dichotomic and Gaussian spins: Critic... Downloads: 2 | |

| A Solvable Mean Field Model of a Gaussian Spin Glass - Adriano Barra We introduce a mean field spin glass model with gaussian distribuited spins and pairwise interactions, whose couplings are drawn randomly from a normal gaussian distribution too. We completely control the main thermodynamical properties of the model (free energy, phase diagram, fluctuations theory) in the whole phase space. In particular we prove that in thermodynamic limit the free energy equals its replica symmetric expression. Downloads: 3 | |

| Organization and evolution of synthetic idiotypic networks - Elena Agliari We introduce a class of weighted graphs whose properties are meant to mimic the topological features of idiotypic networks, namely the interaction networks involving the B-core of the immune system. Each node is endowed with a bit-string representing the idiotypic specificity of the corresponding B cell and a proper distance between any couple of bit-strings provides the coupling strength between the two nodes... Downloads: 2 | |

| On the equivalence of Hopfield Networks and Boltzmann Machines - Adriano Barra A specific type of neural network, the Restricted Boltzmann Machine (RBM), is implemented for classification and feature detection in machine learning. RBM is characterized by separate layers of visible and hidden units, which are able to learn efficiently a generative model of the observed data. We study a "hybrid" version of RBM's, in which hidden units are analog and visible units are binary, and we show that thermodynamics of visible units are equivalent to those of a Hopfield network, in wh... Downloads: 8 | |

| Effective Interactions in Group Competition with Strategic Diffusive Dynamics - Elena Agliari We analyze, on a random graph, a diffusive strategic dynamics with pairwise interactions, where nor Glauber prescription, neither detailed balance hold. We observe numerically that such a dynamics reaches a well defined steady state that fulfills a shift property: the critical temperature of the canonical ferromagnetic phase transition is higher with respect to the expected equilibrium one, known both numerically via Glauber relaxation or Monte Carlo simulations as well as analytically via cavit... Downloads: 7 | |

| Can persistent Epstein-Barr virus infection induce Chronic Fatigue Syndrome as a Pavlov reflex of the immune response? - Elena Agliari Chronic Fatigue Syndrome is a protracted illness condition (lasting even years) appearing with strong flu symptoms and systemic defiances by the immune system. Here, by means of statistical mechanics techniques, we study the most widely accepted picture for its genesis, namely a persistent acute mononucleosis infection, and we show how such infection may drive the immune system toward an out-of-equilibrium metastable state displaying chronic activation of both humoral and cellular responses (a s... Downloads: 3 | |

| A thermodynamical perspective of immune capabilities - Elena Agliari We consider the mutual interactions, via cytokine exchanges, among helper lymphocytes, B lymphocytes and killer lymphocytes, and we model them as a unique system by means of a tripartite network. Each part includes all the different clones of the same lymphatic subpopulation, whose couplings to the others are either excitatory or inhibitory (mirroring elicitation and suppression by cytokine). First of all, we show that this system can be mapped into an associative neural network, where helper ce... Downloads: 3 | |

| New perspectives in the equilibrium statistical mechanics approach to social and economic sciences - Elena Agliari In this work we review some recent development in the mathematical modelling of quantitative sociology by means of statistical mechanics. After a short pedagogical introduction to static and dynamic properties of many body systems, we develop a theory for agents interactions on random graph. Our approach is based on describing a social network as a graph whose nodes represent agents and links between two of them stand for a reciprocal interaction... Downloads: 5 | |

| Analogue neural networks on correlated random graphs - Elena Agliari We consider a generalization of the Hopfield model, where the entries of patterns are Gaussian and diluted. We focus on the high-storage regime and we investigate analytically the topological properties of the emergent network, as well as the thermodynamic properties of the model. We find that, by properly tuning the dilution in the pattern entries, the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the syst... Downloads: 4 | |

| Notes on the p-spin glass studied via Hamilton-Jacobi and Smooth-Cavity techniques - Elena Agliari In these notes, we continue our investigation of classical toy models of disordered statistical mechanics through various techniques recently developed and tested mainly on the paradigmatic SK spin glass. Here we consider the p-spin-glass model with Ising spins and interactions drawn from a normal distribution N[0,1]. After a general presentation of its properties (e.g. self-averaging of the free energy, existence of a suitable thermodynamic limit), we study its equilibrium behavior within the H... Downloads: 9 | |

| Integration indicators in immigration phenomena. A statistical mechanics perspective - Adriano Barra Integration of immigrants is a complex socioeconomic phenomenon considered difficult to describe, understand, and predict. We address the problem of how integration changes with immigration density, and we propose a novel approach to its study guided by a statistical mechanics perspective. More precisely, we focus on studying the dependence of classical integration quantifiers such as the percentage of jobs, temporary and permanent, given to immigrants, mixed marriages, and newborns with parents... Downloads: 15 | |

| Parallel retrieval of correlated patterns - Elena Agliari In this work, we first revise some extensions of the standard Hopfield model in the low storage limit, namely the correlated attractor case and the multitasking case recently introduced by the authors. The former case is based on a modification of the Hebbian prescription, which induces a coupling between consecutive patterns and this effect is tuned by a parameter $a$. In the latter case, dilution is introduced in pattern entries, in such a way that a fraction $d$ of them is blank... Downloads: 2 | |

| Immune networks: multi-tasking capabilities at medium load - Elena Agliari Associative network models featuring multi-tasking properties have been introduced recently and studied in the low load regime, where the number $P$ of simultaneously retrievable patterns scales with the number $N$ of nodes as $P\sim \log N$. In addition to their relevance in artificial intelligence, these models are increasingly important in immunology, where stored patterns represent strategies to fight pathogens and nodes represent lymphocyte clones... Downloads: 3 | |

| Application of a stochastic modeling to evaluate tuberculosis onset in patients treated with tumor necrosis factor inhibitors - Elena Agliari In this manuscript we apply stochastic modeling to investigate the risk of reactivation of latent mycobacterial infections in patients undergoing treatment with tumor necrosis factor inhibitors. First, we review the perspective proposed by one of the authors in a previous work and which consists in predicting the occurrence of reactivation of latent tuberculosis infection or newly acquired tuberculosis during treatment; this is based on variational procedures on a simple set of parameters (e.g... Downloads: 7 | |

| Anergy in self-directed B lymphocytes from a statistical mechanics perspective - Elena Agliari The ability of the adaptive immune system to discriminate between self and non-self mainly stems from the ontogenic clonal-deletion of lymphocytes expressing strong binding affinity with self-peptides. However, some self-directed lymphocytes may evade selection and still be harmless due to a mechanism called clonal anergy. As for B lymphocytes, two major explanations for anergy developed over three decades: according to "Varela theory", it stems from a proper orchestration of the whole B-reperto... Downloads: 5 | |

| Parallel processing in immune networks - Elena Agliari In this work we adopt a statistical mechanics approach to investigate basic, systemic features exhibited by adaptive immune systems. The lymphocyte network made by B-cells and T-cells is modeled by a bipartite spin-glass, where, following biological prescriptions, links connecting B-cells and T-cells are sparse. Interestingly, the dilution performed on links is shown to make the system able to orchestrate parallel strategies to fight several pathogens at the same time; this multitasking capabili... Downloads: 5 | |